LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
FOURTH SEMESTER – APRIL 2007
ST 4501 – DISTRIBUTION THEORY
Date & Time: 24/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
Answer ALL the questions (10×2=20)
- State the properties of a distribution function.
- Define conditional and marginal distributions.
- What is meant by ‘Pair-wise Independence’ for a set of ‘n’ random variables.
- Define Conditional Variance of a random variable X given a r.v. Y=y.
- Define Moment Generating Function(M.G.F.) of a random variable X.
- Examine the validity of the statement “ X is a Binomial variate with mean 10 and standard deviation 4”.
- If X is binomially distributed with parameters n and p, what is the distribution of
- Define ‘Order Statistics’ and give an example.
- State any two properties of Bivariate Normal distribution.
- State central limit theorem.
Answer any FIVE questions (5×8 =40)
- Let (X,Y) have the joint p.d.f. described as follows:
(X,Y) : (1,1) (1,2) (1,3) (2,1) (2,2) (2,3)
f(X,Y) : 2/15 4/15 3/15 1/15 1/15 4/15. Examine if X and Y are independent.
- The joint p.d.f. of X1 and X2 is : f(x1, x2) =.
[a] Find the conditional p.d.f. of X1 given X2=x2.
[b] conditional mean and variance of X1 given X2=x2
- Derive the recurrence relation for the probabilities of Poisson distribution
- Obtain Mode of Binomial distribution
- Obtain Mean deviation about mean of Laplace distribution
- The random variable X follows Uniform distribution over the interval(0,1). Find the
distribution of Y = -2 log X.
- Obtain raw moments of Student’s ‘t’ distribution, Hence fine the Mean and the Variance.
- Let Y1, Y2, Y3, and Y4 denote the order statistics of a random sample of size 4 from a
distribution having p.d.f. f(x) =. Find P (Y3 > ½ ).
Answer any TWO questions only (2×20 =40)
19.[a] The random variables X and Y have the joint p.d.f. f(x, y) =.
Compute Correlation co-efficient between X and Y. 
[b] Let X and Y are two r.v.s with the p.d.f. f(x, y) =.
Examine whether X and Y are stochastically independent. 
20 [a] Derive M.G.F. of Binomial distribution and hence find its mean and variance.
[b] Show that E(Y| X=x) = (n-x) p2/ (1-p1),if (X,Y) has a Trinomial distribution with
parameters n, p1 and p2.
- [a] Prove that Poisson distribution is a limiting case of Binomial distribution, stating the
[b] Let X and Y follow Bivariate Normal distribution with;m1=3 m2=1 s12 =16 s22 =25 and
r = 3/5.
Determine the following probabilities: (i) P[(3<Y<8)|X=7] (ii) P[ (-3<X<3) | Y=4]
- [a[ If X and Y are two independent Gamma variates with parameters m and n respectively.
Let U=X+Y and V= X / (X+Y)
[i] Find the joint p.d.f. of U and V
[ii] Find the Marginal p.d.f.s’ of U and V
[iii] Show that the variables U and V are independent. 
[b] Derive the p.d.f of F-variate with (n1, n2) d.f. 
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